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This question has been studied even for non-commutative associative rings. See, for example Golubchik, I. Z.; Mikhalëv, A. V. Isomorphisms of the general linear group over an associative ring. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1983, no. 3, 61–72. They prove that if the ring $R$ contains $1/2$ (that is $2$ is invertible), then every isomorphism $\phi: GL_n(R)\to GL_n(R)$, $n\ge 3$, is standard on the subgroup $GE_n(R)$ generated by the elementary and diagonal matrices. "Standard" means "generated by an automorphism of $R$.